Integrated Second-Order Rate Law

For a second-order process in one reactant, reciprocal concentration changes linearly with time. The relation is useful for Test second-order kinetic models and estimate concentrations over time.. For [A]₀=0.10 mol/L, k₂=0.50 L·mol⁻¹·s⁻¹ and t=10 s, [A]=1/(10+5)=0.0667 mol/L. Check: [A] decreases while 1/[A] rises linearly. The final check tests a balance, direction or magnitude relevant to this formula. First map each quantity to the actual experiment, then substitute values. The result describes the modelled relationship; it does not automatically establish every mechanism in the real sample.

The symbols mean: [A]_t: concentration at time t (mol/L); [A]_0: initial concentration (mol/L); k: second-order rate constant (L·mol⁻¹·s⁻¹); t: time (s). Follow [A] in mol/L · k₂ in L·mol⁻¹·s⁻¹ · t in s. Concentrations, activities, volumes and times must use compatible scales. A logarithm needs a dimensionless argument, and a balance must include all contributing species. State temperature and standard state whenever they are part of the definition; otherwise the result may refer to a different chemical condition. An amount is not a mass without molar mass, and activity is not an unnormalised concentration. Say whether inputs were measured or taken from a reference.

This integrated form assumes −d[A]/dt=k₂[A]² and a single reactant concentration controls the rate. A bimolecular reaction with unequal reactants may require a different integrated expression. Check this assumption for the actual sample, temperature and phase. Nonideality, mass transport or side reactions can make a precise calculation systematically wrong. Compare the balanced reaction and the activity or concentration scale with the model, then state the assumption that most affects the result. Near model boundaries, check activities and phase state rather than extending a dilute-solution approximation. Different conditions require a fresh model check.

Recalculate the example: For [A]₀=0.10 mol/L, k₂=0.50 L·mol⁻¹·s⁻¹ and t=10 s, [A]=1/(10+5)=0.0667 mol/L. Check: [A] decreases while 1/[A] rises linearly. Check [A] in mol/L · k₂ in L·mol⁻¹·s⁻¹ · t in s and substitute the value back into the relation. Also ask whether its sign, balance or magnitude is chemically plausible. If a check fails, inspect units, coefficients and reference values before blaming rounding. Substitution checks the algebra; comparing assumptions with the experiment checks whether the model is appropriate. For a single reactant, a linear 1/[A]-versus-time plot supports the model; the fitted slope has units L·mol⁻¹·s⁻¹.

Integrated Second-Order Rate Law is related to Geschwindigkeitsgesetz (Reaktionsordnung), Half-Life by Reaction Order, Arrhenius-Gleichung, but answers a distinct question: Test second-order kinetic models and estimate concentrations over time.. Do not confuse it with a relation for a neighboring quantity or a different standard state. Name the target quantity and sample conditions first, then choose the equation containing that quantity. Similar-looking equations may use different stoichiometric bases or validity ranges. The linked entries may supply inputs for another step, but they do not replace this distinction. This keeps the interpretation auditable.